Special values of trigonometric Dirichlet series and Eichler integrals

Abstract

We provide a general theorem for evaluating trigonometric Dirichlet series of the form ${\sum }_{n\ge 1}\frac{f(\pi n\tau )}{n^s}$, where $f$ is an arbitrary product of the elementary trigonometric functions, $\tau$ a real quadratic irrationality and $s$ an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating the series to combinations of derivatives of Eichler integrals and polylogarithms.

17 pages